a bowl of kernels - university of new mexicocrisp/courses/wavelets/fall13/...a bowl of kernels by...

A Bowl of Kernels

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty

December 03, 2013

By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels

Introduction

When we studied Fourier series we improved convergence byconvolving with the Fejer and Poisson kernels on T. The analogousFejer and Poisson kernels on the real line help us improveconvergence of the Fourier integral. In this project we define theFejer kernel,the Poisson kernel, the heat kernel, the Dirichletkernel, and the conjugate Poisson kernel. We study some of theirproperties, applications, and connections with complex variables.


Kernels

Fejer Kernel on R

FRpxq � R

�sinπRx

πRx

2

R ¡ 0 (1)

The Poisson kernel on R

Py pxq � y

πpx2 � y2q y ¡ 0 (2)

The Heat Kernel on R

Htpxq � e�|x|2

4t?4πt

t ¡ 0 (3)


Kernels

The Dirichlet kernel on R

DRpxq �» R

�Re2πixξdξ � sinp2πRxq

πxR ¡ 0 (4)

The Conjugate Poisson kernel on R

Qy pxq � x

πpx2 � y2q y ¡ 0 (5)


Kernels

Poisson Kernel and Conjugate Poisson Kernel solve the Laplaceequation. ∆P � p B2

Bx2 qP � p B2

By2 qP � 0,

∆Q � p B2

Bx2 qQ � p B2

By2 qQ � 0 for px , yq in the upper half-plane

px P R, y ¡ 0q.Proof(Poisson Kernel):

BPBx � y

πp�1qpx2 � y2q�22x

BPBx2

� 8x2y

πpx2 � y2q3 �2y

πpx2 � y2q2BPBy � x2 � y2

πpx2 � y2q2

BPBy2

� �6yx2 � 2y3

πpx2 � y2q3


Kernels

∆P � 8x2y

πpx2 � y2q3 �2y

πpx2 � y2q2 ��6yx2 � 2y3

πpx2 � y2q3

� 2x2y � 2y3 � 2ypx2 � y2qπpx2 � y2q3

� 0

Theorem

Heat Kernel is a solution of the heat equation on the line.p BBt qH � p B2

Bx2 qH


Approximation of The Identity

In class we learned that an Approximation of the Identity on R is afamily tKtutPΛ where the following holds:

1 The functions tKtutPΛ have mean value 1:³8�8 Ktpxqdx � 1

2 The functions are uniformly integrable in t: There is aconstant C ¥ 0 such that

³8�8 |Ktpxq|dx ¤ C for all t P Λ

3 Concentration of mass at the origin: For eachδ ¡ 0 limtÑt0

³|x |¥δ |Ktpxq|dx � 0


Fejer Kernel is an Approximation to the Identity

Recall the Fejer Kernel is defined as,

FRpxq � R

�sinπRx

πRx

2

R ¡ 0 (6)

Mean value 1 : »RR

�sinπRx

πRx

2

dx � 1 (7)

Let u � πRx , dx � duπR Then we have,

1

π

»R

�sin u

u

2

du � 1

π

»R

1 � cos2 u

u2du (8)

� 1

π

»R

p1 � cos uqp1 � cos uqu2

du (9)


Fejer Kernel

Now we can use integration by parts with u1 � 1 � cos u anddv � 1�cos u

πu2 du1

Then we can use the well known identity,³R

1�cos xx2 dx � π and we

have,

� p1 � cos uq �»R

sin u1du1 � p1 � cos uq � cos u � 1 (10)

Uniformly Integrable in t:

There is a constant C ¥ 0 such that³8�8 |R

�sinπRxπRx

�2|dx ¤ C forall R ¡ 0But by 1, we know

³R R�

sinπRxπRx

�2dx � 1 and also R

�sinπRxπRx

�2¥ 0So just let C=1.


Fejer Kernel

Concentration of mass at the origin: For each

δ ¡ 0 limRÑ8

³|x |¥δ |R

�sinπRxπRx

�2|dx � 0.Let δ ¡ 0,We knowR�

sinπRxπRx

�2 ¤ R 1pπRxq2 � 1

Rpπxq2

So we have, »|x |¥δ

R

�sinπRx

πRx

2

¤ 2

» 8δ

1

Rpπxq2 (11)

� 2 limLÑ8

� 1

Rpπq2L �1

Rpπq2δ (12)

� 2

Rpπq2δ (13)

Now we take the limit in R,

limRÑ8

2

Rpπq2δ � 0 (14)


Continuous Functions of Moderate Decrease

Definition

A function f is said to be a continuous function of moderatedecrease if f is continuous and if there are constants A and ε ¡ 0such that|f pxq| ¤ A

p1�|x |1�εqfor all x P R


Theorem

Neither the Poisson kernel nor the Fejer kernel belong to S(R).For each R ¡ 0 and each y ¡ 0 the Fejer kernel FR and thePoisson kernel Py are continuous functions of moderate decreasewith ε � 1. The conjugate Poisson kernel Qy is not a continuousfunction of moderate decrease.

Proof(For Poisson Kernel) Py pxq � yπpx2�y2q

for y ¡ 0

If 0 y 1,

|Py pxq| � | y

πy2pp xy q2 � 1q |

� 1

pπyqp1 � x2

y2 q


If 0 y 1 then

y2 1

1

y2¡ 1

x2

y2¡ x2

x2

y2� 1 ¡ x2 � 1

Then

|Py pxq| � 1

pπyqp1 � x2

y2 q

1

πypx2 � 1q


If y ¥ 1

y ¥ 1

y2 ¥ 1

x2 � y2gleqx2 � 1

Then

|Py pxq| � | y

πy2pp xy q2 � 1q |

¤ y

πpx2 � 1qHence,

Ay �#

1πy if 0 y 1,yπ if y ¥ 1


Kernels

Theorem

Let SR f denote the partial Fourier integral of f, defined by

SR f pxq �» R

�Rf pξqe2πixξdξ (15)

Then SR f pxq � DR � f and FRpxq � 1R

³R0 Dtpxqdt.

Proof:

DR � f �»RDRpx � yqf pyqdy (16)

�»R

» R

�Re2πpx�yqξdξf pyqdy (17)

�» R

�R

»Re�2πiyξf pyqdye2πixξdξ �

» R

�R

ˆf pξqe2πixξdξ

(18)

(19)By Nuriye Atasever, Cesar Alvarado, and Patrick Doherty A Bowl of Kernels

Kernels

Also, FRpxq � 1R

³R0 Dtpxqdt because,

1

R

» R

0Dtpxqdt � 1

R

» R

0

sinp2πtxqπx

dt (20)

� 1

R

��1

πx

cosp2πtxq2πx

��R0

(21)

� 1

R2π2x2pcosp0q � cosp2πRxqq (22)

� psinpπRxq2Rπ2x2

� FRpxq (23)


Integral Cesaro Means

We have the Fejer Kernel can be written as the integral mean ofthe Dirichlet kernel, and the Fejer kernel define an approximationto the identity on R as R Ñ8. Therefore, the integral Cesaromeans of a function of moderate decrease converge to f as R Ñ8Theorem

σR f pxq � 1

R

» R

0St f pxqdt � FR � f pxq Ñ f pxq (24)

The convergence is both uniform and in Lp



We will prove for p=1.We will show,

limRÑ8

||FR � f � f ||1 � 0 (25)

|FRpxq � f pxq � f pxq| ��»

Rf px � yqFRpyqdy � f pxq

�� (26)

��»

Rpf px � yq � f pxqqFRpyqdy

�� (27)

¤»R|pf px � yq � f pxqq| |FRpyq| dy (28)

||FR � f � f ||1 ¤»R

»R|pf px � yq � f pxqq| |FRpyq| dydx (29)

(30)



By Fubini,

||FR � f � f ||1 ¤»R||pf px � yq � f pxqq| |1 |FRpyq| dy (31)

�»|y |¡δ

||pf px � yq � f pxqq| |1 |FRpyq| dy (32)

�»|y | δ

||pf px � yq � f pxqq| |1 |FRpyq| dy (33)

� I1 � I2 (34)

Notice, I2 �³|y | δ ||pf px � yq � f pxqq| |1 |FRpyq| dy can be made

arbitrarily small by continuity of f.



Also, I1 �³|y |¡δ ||pf px � yq � f pxqq| |1 |FRpyq| dy can be made

arbitrarily small since FR is an approximation to the identity. Thisconcludes the proof, and we have

σR f pxq � 1

R

» R

0St f pxqdt � FR � f pxq Ñ f pxq (35)


Fourier Transform of the Fejer Kernel

Since the Fejer Kernel is a continuous function of moderatedecrease, we can actualy calculate its Fourier Transform.Let f be any function with a Fourier Transform supported on R andf pξq � 0

FR � f � 1

R

» R

0Dtpxqdt � f (36)

� 1

R

» R

0pDtpxq � f qdt (37)

� 1

R

» R

0

�» t

�tf pξqe2πixξdξ

dt (38)

�» R

�Rf pξqe2πixξ

�» R

|ξ|

1

Rdt

�dξ (39)


Fourier Transform of the Fejer Kernel

FR � f �» R

�Rf pξqe2πixξ

�1 � |ξ|

R

dξ (40)

�» 8�8

f pξq�

1 � |ξ|R

χr�R,Rspξqe2πixξdξ (41)

��f pξq

�1 � |ξ|

R

χr�R,Rspξq

q (42)

Now we can take the transform,

{FR � f � f pξq�

1 � |ξ|R

χr�R,Rspξq (43)

{FRpξq ˆf pξq � f pξq�

1 � |ξ|R

χr�R,Rspξq (44)

{FRpξq � �1 � |ξ|R

χr�R,Rspξq (45)


Connection with Complex Variables

Consider a real valued function f P L2pRq. Let F pz) be twice theanalytic extension of f to the upper half planeR2� � tz � x � iy : y ¡ 0u Then F pzq is given explicitly by the

well known Cauchy Integral Formula:

F pzq � 1

πi

»R

f ptqt � z

dt (46)

If we separate F pzq into its real and imaginary parts, we get

F pzq � 1

πi

»R

f ptqpt � xq � iy

dt (47)



F pzq � 1

πi

»R

f ptqppt � xq � iyqpt � xq2 � y2

dt (48)

� i

π

»R

f ptqpx � tqpt � xq2 � y2

dt � 1

π

»R

f ptqypt � xq2 � y2

dt (49)

� f � Py pxq � ipf � Qy pxqq (50)

� u � iv (51)

The function u is called the harmonic extension of f to the upperhalf plane , while v is called the harmonic conjugate of u.


Theorem

Qy R L1pRq, but Qy P L2pRqProof

»R| x

πpx2 � y2q |dx �» 0

�8

�xπpx2 � y2qdx �

» 80

x

πpx2 � y2q

� �» y2

8

1

2π

1

udu �

» 8y2

1

2π

1

udu

� 1

2π

» 8y2

1

udu �

» 8y2

1

2π

1

udu

� 1

π

» 8y2

1

udu

� 1

πln|u|8y2

� 8So Qy pxq R L1pRq



The conjugate Poisson Kernel is not in L1pRq and is not a functionof moderate decrease, so we define its Fourier transform as aprincipal value:

ˆQy pξq � limRÑ8

» R

�Re�2πiξxQy pxqdx (52)

� limRÑ8

» R

�Re�2πiξx x

πpx2 � y2qdx (53)

(54)

Now, consider the function f pzq � e�2πiξz zπpz2�y2q

and apply

residue theorem for a well chosen contour in the complex plane.


connection to complex variables

For ξ 0 we choose the semi circle in the upper half plane and forξ ¡ 0 we choose the semi circle in the lower half plane. Now, letsfind the residues of f at iy and �iy .

Respf pzq, iyq � e�2πiξpiyq

2π(55)

� e2πiξy

2πfor ξ o (56)

Respf pzq,�iyq � e�2πiξp�iyq

2π(57)

� e�2πiξy

2πfor ξ ¡ 0 (58)


Connection with complex variables

Therefore

ˆQy pξq � �isgnpξqe�2πy |ξ| (59)

Definition

The Hilbert Transform H is defined by :Hf pxq � 1

π limεÑ0

³|x�y |¡ε

f pyqx�y dy for f P L2pRq

Definition

Hf pξq � �isgnpξqf pξq

We can see that the Hilbert transform can be written as theprincipal value of the convolution of f and 1

πx . It is the response tof of a linear time invariant filter(Hilbert Transformer) havingimpulse response 1

πx



The Hilbert transform find a harmonic conjugate yptq for a realfunction xptq so that zptq � xptq � iyptq can be analyticallyextended from R to the upper half plane.In signal processing, theHT can be interpreted as a way to represent a narrow band signalin terms of amplitude and frequency modulation.In the Fourierside, we can think of the HT as a phase shifter by 90 degrees. Ifwe take the limit of ˆQy pξq as y goes to zero, we get

ˆQy pξq Ñ �isgnpξq (60)

Then as y Ñ 0, Qy pxq � f approaches the Hilbert transform Hf inL2pRq


Applications

Consider the initial value problem for the heat equation:ut � uxx � 0 in Rxp0,8q with u � g on Rxpt � 0qIf we take the Fourier transform of u in x we get: ut � y2u � 0 fort ¡ 0 and u � g for t � 0 Consequently, u � e�ty2

g and

u � �e�ty2

g therefore u � g�F

p2πq12

where F � e�ty2. But then

F � �pe�ty2q (61)

� 1

p2πq 12

»Re ixy�ty2

dy (62)

� 1

2te�x2

4t (63)

Therefore,

upx , tq � 1

p4πtq 12

»Re�|x�y |2gpyqdy x P R, t ¡ 0 (64)


References

Harmonic Analysis:from Fourier to Wavelets. Cristina Pereyraand Lesly Ward. AMS, 2010.

Partial Differential Equations. Lawrance Evans.AMS, 1998.

Fourier Analysis. Stein and Shakarchi.Princetton UniversityPress, 2003.


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